Analysis and Graphing
Our Learning objective: Is to explore and explain why
the denominator of a rational function cannot be zero. Thus
recognizing these values as the places where vertical
asymptotes occur, (which are disastrous things to have), and
graphically what vertical asymptotes look like and mean.
What is a Rational Function?
• A rational function has the form of a
polynomial over a polynomial.
• The bottom polynomial must never be zero!
• If the bottom polynomial is zero this will make
the function undefined.
• Hence, these values
are left out of the function’s
domain. Equations of
Self Check 1
Which function below is a rational function?
What are Vertical Asymptotes?
• Vertical asymptotes are the values that make the
denominator go to zero, which makes the
• These values are represented by x = a and/or
x = b, where a and b are real numbers.
• What do you notice about the equations x = a
and x = b?
• They are the equations of
vertical lines, hence the
name vertical asymptotes! Basic Rational
How do we find Vertical Asymptotes?
• We have to find the places where the
denominator goes to zero.
• We do this by setting the denominator (bottom
polynomial) equal to zero and finding the
values of x that make it zero.
• This is when we get the
equations of the form x = a. Vertical line
at x = a.
Self Check 2
• What are the vertical asymptotes for the
Do Vertical Asymptotes Always Exist?
• No. If the zeros of the bottom polynomial are
complex numbers, then the function does not
have vertical asymptotes.
• See the diagram to the right?
This rational function’s
denominator does not go to
zero. We can tell because the ends of the
graph get close to the x – axis but do not cross
the x- axis.
Self Check 3
• Which of the following functions has vertical
asymptotes? Set each denominator equal to
zero and solve for the values of x.
How can we Identify Vertical
• Vertical asymptotes are identified as dashed
or dotted vertical lines in the plane.
• You guessed it!! The equations of those
vertical lines are the values of x that make the
denominator equal to zero or x = a and x = b.
What do you notice about the
graph of the function as it
approaches the vertical line that
is the vertical asymptote?
Self Check 4
Which diagram illustrates a vertical asymptote?
What do Vertical Asymptotes mean to
the graph of the function?
• Since the vertical asymptotes make the
function undefined, the graph of the function
NEVER crosses or touches the vertical
• Hence, the graph bends and diverges to
positive infinity or negative infinity on each
side of each vertical asymptote!
• See the next slide for two examples!!
Graphs of Functions Containing
Rational Function with one This rational function has
vertical asymptote. It two vertical asymptotes. So
diverges, bends toward you have to determine
positive infinity on the right which way it diverges on
and bends toward negative each side of each vertical
infinity on the left. asymptote. Here that is
four different calculations!!
What is the most vertical asymptotes a function
can have? The tangent function has infinite!!
Self Check 5
Tell which way the function diverges as it
approaches the vertical asymptote from the
right and the left.
Are Vertical Asymptotes Good?
• What do you think it means to say the graph bends or
diverges toward positive or negative infinity?
• If you are analyzing the cost of producing something
and the model’s graph bends toward positive infinity,
do you think that is something good?
• Vertical asymptotes are disastrous things, because
when the function diverges it means the function
forever goes in the direction of infinity. Thus we call
the function undefined.
• We avoid models that behave in this way because they
are unstable and can have disastrous effects.
Self Check Answers
2. There are two: x = - 1/2 and x = 1/2
5. From the left goes toward negative infinity
and from the right toward positive infinity.